How To Find Instantaneous Velocity From A Position Time Graph

Learning Objectives

By the cease of this section, you will be able to:

• Explain the difference between boilerplate velocity and instantaneous velocity.
• Describe the departure between velocity and speed.
• Calculate the instantaneous velocity given the mathematical equation for the velocity.
• Calculate the speed given the instantaneous velocity.

Nosotros have now seen how to calculate the boilerplate velocity between two positions. However, since objects in the real world motility continuously through space and time, nosotros would similar to find the velocity of an object at any single point. We can find the velocity of the object anywhere along its path past using some fundamental principles of calculus. This section gives us better insight into the physics of motion and will be useful in later capacity.

Instantaneous Velocity

The quantity that tells us how fast an object is moving anywhere along its path is the instantaneous velocity, unremarkably called simply velocity. It is the average velocity between ii points on the path in the limit that the time (and therefore the displacement) between the 2 points approaches null. To illustrate this thought mathematically, we demand to express position x every bit a continuous function of t denoted by x(t). The expression for the average velocity between two points using this notation is [latex] \overset{\text{–}}{v}=\frac{ten({t}_{2})-x({t}_{1})}{{t}_{2}-{t}_{1}} [/latex]. To discover the instantaneous velocity at any position, we let [latex] {t}_{1}=t [/latex] and [latex] {t}_{2}=t+\text{Δ}t [/latex]. After inserting these expressions into the equation for the average velocity and taking the limit every bit [latex] \text{Δ}t\to 0 [/latex], we observe the expression for the instantaneous velocity:

[latex] 5(t)=\underset{\text{Δ}t\to 0}{\text{lim}}\frac{x(t+\text{Δ}t)-10(t)}{\text{Δ}t}=\frac{dx(t)}{dt}. [/latex]

Instantaneous Velocity

The instantaneous velocity of an object is the limit of the average velocity as the elapsed time approaches zippo, or the derivative of x with respect to t:

[latex] v(t)=\frac{d}{dt}x(t). [/latex]

Similar average velocity, instantaneous velocity is a vector with dimension of length per fourth dimension. The instantaneous velocity at a specific time point [latex] {t}_{0} [/latex] is the rate of change of the position function, which is the slope of the position office [latex] ten(t) [/latex] at [latex] {t}_{0} [/latex]. (Figure) shows how the average velocity [latex] \overset{\text{–}}{v}=\frac{\text{Δ}x}{\text{Δ}t} [/latex] between 2 times approaches the instantaneous velocity at [latex] {t}_{0}. [/latex] The instantaneous velocity is shown at time [latex] {t}_{0} [/latex], which happens to be at the maximum of the position function. The gradient of the position graph is nada at this point, and thus the instantaneous velocity is cypher. At other times, [latex] {t}_{i},{t}_{ii} [/latex], and then on, the instantaneous velocity is not aught because the slope of the position graph would be positive or negative. If the position role had a minimum, the slope of the position graph would as well exist zip, giving an instantaneous velocity of zero there as well. Thus, the zeros of the velocity role give the minimum and maximum of the position office.

Effigy 3.6 In a graph of position versus time, the instantaneous velocity is the gradient of the tangent line at a given betoken. The average velocities [latex] \overset{\text{–}}{5}=\frac{\text{Δ}10}{\text{Δ}t}=\frac{{x}_{\text{f}}-{x}_{\text{i}}}{{t}_{\text{f}}-{t}_{\text{i}}} [/latex] between times [latex] \text{Δ}t={t}_{6}-{t}_{one},\text{Δ}t={t}_{5}-{t}_{two},\text{and}\,\text{Δ}t={t}_{iv}-{t}_{iii} [/latex] are shown. When [latex] \text{Δ}t\to 0 [/latex], the boilerplate velocity approaches the instantaneous velocity at [latex] t={t}_{0} [/latex].

Example

Finding Velocity from a Position-Versus-Time Graph

Given the position-versus-time graph of (Figure), discover the velocity-versus-time graph.

Figure iii.7 The object starts out in the positive management, stops for a short time, and then reverses direction, heading back toward the origin. Discover that the object comes to rest instantaneously, which would crave an infinite forcefulness. Thus, the graph is an approximation of motion in the real world. (The concept of forcefulness is discussed in Newton'south Laws of Motion.)

Strategy

The graph contains three straight lines during 3 time intervals. We find the velocity during each fourth dimension interval past taking the slope of the line using the filigree.

Solution

Significance

During the time interval between 0 southward and 0.v s, the object's position is moving away from the origin and the position-versus-fourth dimension curve has a positive slope. At any point along the curve during this time interval, we can find the instantaneous velocity by taking its slope, which is +1 grand/south, equally shown in (Figure). In the subsequent time interval, between 0.v due south and 1.0 southward, the position doesn't change and nosotros see the slope is cipher. From 1.0 s to 2.0 s, the object is moving back toward the origin and the slope is −0.5 m/due south. The object has reversed management and has a negative velocity.

Speed

In everyday linguistic communication, well-nigh people use the terms speed and velocity interchangeably. In physics, notwithstanding, they do not have the same pregnant and are distinct concepts. One major deviation is that speed has no direction; that is, speed is a scalar.

Nosotros can calculate the average speed by finding the total distance traveled divided by the elapsed time:

[latex] \text{Average speed}=\overset{\text{–}}{southward}=\frac{\text{Total distance}}{\text{Elapsed time}}. [/latex]

Boilerplate speed is not necessarily the aforementioned every bit the magnitude of the average velocity, which is constitute by dividing the magnitude of the total deportation by the elapsed time. For instance, if a trip starts and ends at the same location, the total displacement is zero, and therefore the boilerplate velocity is goose egg. The average speed, however, is non nil, considering the total altitude traveled is greater than zip. If nosotros take a road trip of 300 km and demand to be at our destination at a certain fourth dimension, then we would be interested in our average speed.

However, we tin summate the instantaneous speed from the magnitude of the instantaneous velocity:

[latex] \text{Instantaneous speed}=|five(t)|. [/latex]

If a particle is moving along the ten-centrality at +7.0 m/s and another particle is moving along the same axis at −seven.0 m/south, they have different velocities, but both have the aforementioned speed of 7.0 m/due south. Some typical speeds are shown in the following tabular array.

Speeds of Diverse Objects*Escape velocity is the velocity at which an object must be launched so that it overcomes Globe's gravity and is not pulled dorsum toward Globe.

Speed	m/s	mi/h
Continental drift	[latex] {10}^{-seven} [/latex]	[latex] 2\,×\,{10}^{-seven} [/latex]
Brisk walk	1.7	3.9
Cyclist	4.iv	10
Dart runner	12.two	27
Rural speed limit	24.half dozen	56
Official country speed record	341.1	763
Speed of sound at sea level	343	768
Space shuttle on reentry	7800	17,500
Escape velocity of World*	11,200	25,000
Orbital speed of Earth around the Sunday	29,783	66,623
Speed of light in a vacuum	299,792,458	670,616,629

Calculating Instantaneous Velocity

When calculating instantaneous velocity, we need to specify the explicit form of the position function x(t). For the moment, let'due south utilise polynomials [latex] x(t)=A{t}^{n} [/latex], because they are easily differentiated using the ability rule of calculus:

[latex] \frac{dx(t)}{dt}=nA{t}^{n-i}. [/latex]

The following example illustrates the use of (Figure).

Instance

Instantaneous Velocity Versus Average Velocity

The position of a particle is given by [latex] x(t)=three.0t+0.5{t}^{3}\,\text{g} [/latex].

1. Using (Figure) and (Figure), discover the instantaneous velocity at [latex] t=2.0 [/latex] s.
2. Calculate the average velocity between 1.0 southward and three.0 south.

Strategy(Figure) gives the instantaneous velocity of the particle every bit the derivative of the position part. Looking at the course of the position function given, we encounter that information technology is a polynomial in t. Therefore, we can use (Effigy), the power rule from calculus, to find the solution. Nosotros employ (Figure) to summate the average velocity of the particle.

Solution

1. [latex] 5(t)=\frac{dx(t)}{dt}=iii.0+1.v{t}^{2}\,\text{m/s} [/latex].Substituting t = 2.0 due south into this equation gives [latex] v(two.0\,\text{s})=[3.0+1.5{(2.0)}^{2}]\,\text{thou/s}=ix.0\,\text{m/due south} [/latex].
2. To determine the average velocity of the particle between 1.0 s and 3.0 s, we calculate the values of ten(1.0 due south) and x(iii.0 s):

   [latex] x(1.0\,\text{s})=[(3.0)(one.0)+0.five{(1.0)}^{3}]\,\text{m}=3.5\,\text{k} [/latex]

   [latex] x(3.0\,\text{s})=[(three.0)(3.0)+0.5{(3.0)}^{3}]\,\text{m}=22.5\,\text{thou.} [/latex]

   Then the average velocity is

   [latex] \overset{\text{–}}{v}=\frac{ten(3.0\,\text{due south})-ten(1.0\,\text{due south})}{t(3.0\,\text{s})-t(one.0\,\text{s})}=\frac{22.5-3.5\,\text{m}}{three.0-ane.0\,\text{s}}=9.5\,\text{m/s}\text{.} [/latex]

Significance

In the limit that the time interval used to calculate [latex] \overset{\text{−}}{5} [/latex] goes to nada, the value obtained for [latex] \overset{\text{−}}{5} [/latex] converges to the value of 5.

Example

Instantaneous Velocity Versus Speed

Consider the move of a particle in which the position is [latex] x(t)=three.0t-three{t}^{2}\,\text{m} [/latex].

1. What is the instantaneous velocity at t = 0.25 s, t = 0.50 s, and t = 1.0 due south?
2. What is the speed of the particle at these times?

Strategy

The instantaneous velocity is the derivative of the position function and the speed is the magnitude of the instantaneous velocity. Nosotros utilize (Figure) and (Figure) to solve for instantaneous velocity.

Solution

1. Bear witness Reply

   [latex] v(t)=\frac{dx(t)}{dt}=iii.0-6.0t\,\text{one thousand/s} [/latex]

2. Prove Respond

   [latex] v(0.25\,\text{s})=ane.50\,\text{m/south,}5(0.5\,\text{s})=0\,\text{m/s,}v(1.0\,\text{s})=-three.0\,\text{g/southward} [/latex]

3. Show Answer

   [latex] \text{Speed}=|v(t)|=1.fifty\,\text{m/s},0.0\,\text{m/south,}\,\text{and}\,iii.0\,\text{m/s} [/latex]

Significance

The velocity of the particle gives united states of america management data, indicating the particle is moving to the left (w) or right (east). The speed gives the magnitude of the velocity. By graphing the position, velocity, and speed as functions of time, we tin can understand these concepts visually (Figure). In (a), the graph shows the particle moving in the positive management until t = 0.5 s, when it reverses direction. The reversal of direction can also exist seen in (b) at 0.5 due south where the velocity is zero and so turns negative. At ane.0 s it is back at the origin where it started. The particle's velocity at 1.0 s in (b) is negative, because it is traveling in the negative direction. Merely in (c), all the same, its speed is positive and remains positive throughout the travel fourth dimension. Nosotros tin besides interpret velocity as the slope of the position-versus-fourth dimension graph. The slope of 10(t) is decreasing toward zero, condign zero at 0.5 southward and increasingly negative thereafter. This analysis of comparing the graphs of position, velocity, and speed helps catch errors in calculations. The graphs must be consistent with each other and aid translate the calculations.

Figure iii.9 (a) Position: 10(t) versus time. (b) Velocity: v(t) versus fourth dimension. The slope of the position graph is the velocity. A crude comparison of the slopes of the tangent lines in (a) at 0.25 southward, 0.5 due south, and one.0 s with the values for velocity at the corresponding times indicates they are the same values. (c) Speed: [latex] |v(t)| [/latex] versus fourth dimension. Speed is always a positive number.

Check Your Agreement

The position of an object as a function of time is [latex] x(t)=-iii{t}^{two}\,\text{m} [/latex]. (a) What is the velocity of the object as a function of fourth dimension? (b) Is the velocity ever positive? (c) What are the velocity and speed at t = 1.0 s?

Evidence Solution

(a) Taking the derivative of x(t) gives five(t) = −6t k/s. (b) No, because fourth dimension tin never be negative. (c) The velocity is v(ane.0 s) = −half-dozen 1000/s and the speed is [latex] |v(1.0\,\text{s})|=half dozen\,\text{m/s} [/latex].

Summary

• Instantaneous velocity is a continuous function of time and gives the velocity at whatsoever point in time during a particle'south move. We tin can calculate the instantaneous velocity at a specific fourth dimension by taking the derivative of the position function, which gives us the functional form of instantaneous velocity 5(t).
• Instantaneous velocity is a vector and can be negative.
• Instantaneous speed is institute by taking the absolute value of instantaneous velocity, and it is always positive.
• Average speed is total distance traveled divided by elapsed time.
• The slope of a position-versus-time graph at a specific time gives instantaneous velocity at that time.

Conceptual Questions

There is a distinction betwixt average speed and the magnitude of average velocity. Give an example that illustrates the difference betwixt these two quantities.

Show Solution

Average speed is the total distance traveled divided past the elapsed time. If you become for a walk, leaving and returning to your home, your average speed is a positive number. Since Average velocity = Displacement/Elapsed time, your average velocity is zippo.

Does the speedometer of a car measure speed or velocity?

If y'all divide the total altitude traveled on a motorcar trip (as adamant by the odometer) past the elapsed fourth dimension of the trip, are you calculating average speed or magnitude of average velocity? Under what circumstances are these two quantities the same?

Show Solution

Average speed. They are the aforementioned if the car doesn't reverse management.

How are instantaneous velocity and instantaneous speed related to i another? How do they differ?

Problems

A woodchuck runs 20 chiliad to the right in 5 south, then turns and runs 10 m to the left in iii s. (a) What is the average velocity of the woodchuck? (b) What is its average speed?

Sketch the velocity-versus-time graph from the following position-versus-time graph.

Show Answer

Sketch the velocity-versus-time graph from the following position-versus-time graph.

Given the post-obit velocity-versus-time graph, sketch the position-versus-fourth dimension graph.

Testify Answer

An object has a position part x(t) = 5t m. (a) What is the velocity as a function of time? (b) Graph the position function and the velocity function.

A particle moves along the ten-axis according to [latex] x(t)=10t-2{t}^{2}\,\text{m} [/latex]. (a) What is the instantaneous velocity at t = two south and t = 3 southward? (b) What is the instantaneous speed at these times? (c) What is the average velocity between t = 2 southward and t = three south?

Show Respond

a. [latex] v(t)=(10-4t)\text{g/south} [/latex]; 5(ii south) = ii chiliad/southward, five(iii s) = −2 thousand/s; b. [latex] |v(2\,\text{s})|=2\,\text{grand/s},|v(3\,\text{s})|=2\,\text{m/s} [/latex]; (c) [latex] \overset{\text{–}}{v}=0\,\text{m/s} [/latex]

Unreasonable results. A particle moves along the 10-centrality according to [latex] 10(t)=3{t}^{iii}+5t\text{​} [/latex]. At what time is the velocity of the particle equal to nada? Is this reasonable?

Glossary

instantaneous velocity

the velocity at a specific instant or time point

instantaneous speed

the accented value of the instantaneous velocity

boilerplate speed

the full distance traveled divided by elapsed time

Source: https://courses.lumenlearning.com/suny-osuniversityphysics/chapter/3-2-instantaneous-velocity-and-speed/

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